What 1986 breakthrough finally taught neural networks with many layers how to learn?

The answer

In 1986, David Rumelhart, Geoffrey Hinton, and Ronald Williams published 'Learning representations by back-propagating errors' in Nature, demonstrating that backpropagation could effectively train multi-layer neural networks. The algorithm works by computing the gradient of the loss function with respect to each weight, propagating error signals backward from the output layer through hidden layers. Although backpropagation had been independently discovered several times before - by Paul Werbos in 1974, and by David Parker and Yann LeCun in 1985 - the Rumelhart-Hinton-Williams paper provided clear experimental results showing that hidden layers could learn useful internal representations, reigniting interest in neural networks after nearly two decades of neglect following the Perceptrons critique.